01 -
In a large skillet over medium-high heat, melt 2 tablespoons butter. Add onions and season with salt and pepper. Cover and cook, stirring occasionally, until softened, about 3 to 5 minutes. Uncover, reduce heat to medium-low, and continue cooking, stirring occasionally, until deep golden brown, 20 to 25 minutes. If onions begin to dry out or stick, add a few tablespoons of water at a time, scraping up browned bits. If desired, deglaze skillet with vinegar and cook until liquid evaporates, about 1 minute. Transfer onions to a medium bowl, adjust seasoning with salt and pepper, and set aside. Wipe out skillet if necessary.
02 -
Add grated cheese to the caramelized onions and mix to combine. Place two slices of bread on a work surface and divide the cheese-onion mixture evenly between them. Top with the remaining bread slices and press down gently.
03 -
In the same skillet, melt 1 tablespoon butter over medium heat. Add the sandwiches and cook until the bottoms turn golden brown, 3 to 4 minutes. Adjust heat as needed to avoid over-browning. Add the remaining 1 tablespoon butter, flip the sandwiches, press down gently, and cook until cheese is fully melted and the other side is golden brown, 3 to 4 minutes. Serve immediately.